This is a fairly simple and self explanatory mini-game. Enter a Qu's Marsh with Quina in your party, then make your way to the middle of the marsh where the frogs dwell and Quina will ask if she can catch frogs. Saying "Yes" will start a mini-game where you control Quina and can catch frogs by pressing X when you get close enough to them on land. Frogs can not be caught if they are in the water. Try hanging out a little away from the fogs and when one hops onto land run over qucikly mashing the X button. Catching 99 different frogs will earn you this trophy.
There are four different Qu's Marshes in Gaia. The first one is located outside of Lindblum and is unlocked for you during the main story. The second is in the Outer Continent between Conde Petie and the Black Mage Village. The third is on the Forgotten Continent, located north of where Kuja drops you off on your way to Oeilvert. The final one is on an island to the South of the Forgotten Continent and can only be accessed with the Blue Narciss, the Hilda Garde 3 or when Choco learns how to fly.
To speed up the frog regeneration process, it is best to always leave one male and one female frog in the marsh before leaving. You can easily differentiate between male and female frogs by looking at their feet. Male frogs have yellow feet, female frogs have red feet. By leaving one male and one female, you will ensure that the marsh will be repopulated quickly. If you do catch all frogs in the marsh, exiting and re-entering will cause two tadpoles to spawn.
Frogs start as tadpoles and cannot be caught. Once enough time has passed, the tadpoles become baby frogs and can be caught but do not reproduce and are genderless. Only adult frogs reproduce. Thus, don't be afraid to catch baby frogs if there is already an adult male and adult female frog in the marsh.
Additionally, the presence of a golden frog speeds up the frog maturation process dramatically. There can only be one golden frog present in the game so whichever marsh it is in will have fast growing frogs. I would not recommend catching the golden frog unless a) you still haven't got the Going for the Gold trophy or b) you want it to respawn in a different marsh. Note that you can’t choose which marsh it will respawn in.
Depending on the marsh and the presence of a golden frog, it can take around two to three hours to fully respawn a marsh's frogs. It is recommended to visit all four marshes in one go, waiting a couple of hours so that the frogs respawn, then doing another round of marsh hopping and so on. Be advised that visiting a marsh during the regeneration process will stunt the frogs' growth so if you want to be safe, save before entering the marsh and reload your save if there are less frogs than you want.
Also be advised that there is a boss fight immediately after you catch the 99th frog so saving after the 98th frog is advisable.
Once you've caught 99 frogs, the trophy will pop.